There are many things called log-something, related to work of Fontaine, Illusie, Kato and many others. See for example this article.
I know nothing about this, but many log-something cohomology theories seem to arise.
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AG (Algebraic geometry)
Log
Illusie talk: A classical theorem, due to Deligne (1969), asserts that if Y is a scheme of characteristic zero and f : X ! Y is a proper and smooth morphism, then the relative Hodge to de Rham spectral sequence of f degenerates at E1 and its initial term is locally free of finite type. I will discuss logarithmic generalizations of this result, using Kummer ´étale sites and Kato-Nakayama spaces. I will sketch possible further developments. This is joint work with K. Kato and C. Nakayama.
nLab page on Logarithmic cohomology